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Your info is somewhat awry

Marriages Mar 1873 (>99%)
GILL Mary Ruth Newport Mon 11a 231

Where did you look for the woman's death?

.................this is on freebmd

Deaths Jun 1888 (>99%)
DAWSON Mary Ruth 39 Birmingham 6d 2

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Look for info about deaths on freebmd

And please clarify your request

You have 'born 1846' in your title...and 'married 1843' in the body of your post

Don't assume that a re-marriage was necessarily preceded by a death

A ships' chandler, he married first Mary RUTH Gill, daughter of Robert Gill, and Eleanor Gill (born Thompson), in the 1st quarter of 1873 in Newport, Monmouthshire. Children include Eleanor Jane, Gwendoline Maud, Olive Ruth, Charles Ernest and Thomas Birbeck.
In 1881 they were living at a house called Rothbury in Stow Park Newport. Mary Ruth's mother lived next door.
In the following decade Mary Ruth must have died. This would have been between 1886 (Olive's birth year) and 1889. Unfortunately I can find no death record and have no idea of the circumstances.
Sometime in this decade Eleanor Gill (Thomas' mother-in-law) moved into Rothbury, where she remained until her death in 1898. She left a will.
Also Thomas remarried. He married Mary ANN Powell on 10/12/1889 in Newport.
They had two children but one died.
The child who survived, Herbert Percy Dawson, was baptised, St Woolos 17/02/1891
I am trying to find where and when Mary RUTH Dawson died and, should anyone happen to have the certificate, the circumstances? A long shot I know!
I am also trying to find a date and the place of Thomas Dawson's death and if he left a will?
Thanks for your time
Carole